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<div style="margin-left: 40px;"><span style="font-weight: bold;">&nbsp;2.&nbsp;
	<a name="." href="#Dynamic_models">Dynamic models</a><a name=".">.</a></span><br>
<div style="margin-left: 40px;">2.1. <a href="#System_Models">System Models</a><br>
<div style="margin-left: 40px;">2.1.1. <a href="#Furutas_pendulum_">Furuta's 
	pendulum</a><br>
	2.1.2. <a href="#Cartpole">Cartpole</a><br>
	2.1.3. <a href="#Acrobot_and_Pendubot">Acrobot and Pendubot</a><br>
	2.1.4. <a href="#Inertia_Wheel">Inertia Wheel</a><br>
	<p>&nbsp;</div>
</div>
</div>
<hr noshade>
<blockquote>
	<p><b><a name="Dynamic_models">2. Dynamic models</a></b></p>
	<p align="justify">This section introduces the complete mathematical 
	framework used to model the embedded dynamics models for each underactuated 
	system. The aim of this help guide consists in<br>
	the description of the dynamic model of the system and its final 
	representation in variable state form, in order to provide to the user&nbsp; 
	the necessary foundations to design controllers <br>
	based on this model.<br>
&nbsp;</p>
	<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr noshade>
</blockquote>
<p align="left">&nbsp;</p>
<blockquote>
	<blockquote>
		<p><b><a name="System_Models">2.1. System Models</a></b></p>
		<p align="justify">Five underactuated models have been embedded into 
		VISUNS. The mathematical frameworks included into the platform, have 
		been tested and validated&nbsp; with&nbsp; other<br>
		implementations (references are included at the end of the mathematical 
		description) and with several proofs using Matlab-Simulink. Inside the 
		equations of motion for<br>
		each system, Newton-Euler and Lagrange formulations have been used. The 
		non-linear dynamics models contains the major physics&nbsp; variables&nbsp; 
		as possible in order to <br>
		obtain and better approach of the response of the simulated system (The 
		user may modify the dynamics and kinematics parameters).<br>
&nbsp;</p>
		<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
		<blockquote>
			<p align="justify"><b><a name="Furutas_pendulum_">2.1.1. Furuta's pendulum</a></b></p>
			<p align="center">&nbsp;<img border="0" src="images/furuta.JPG" width="399" height="312"></p>
			<p align="justify"><span style="font-family: Times New Roman">For 
			the Furuta's pendulum nonlinear model, Euler-Lagrange formulations 
			have been established, in order to obtain the Lagrangian based on 
			the difference between the</span><br>
&nbsp;<span style="font-family: Times New Roman">kinetic <i>(K)</i> and 
			potential <i>(U)</i> energy of the system. The lagrangian <i>(L)</i> 
			is defined as:</span></p>
			<p align="justify">
			<img border="0" src="images/furutadyn.JPG" width="867" height="538"></p>
			<p align="justify"><span style="font-family: Times New Roman">&nbsp;</span><span style="font-family: Times New Roman; font-style: italic">Parameters description:</span><br>
			<br>
			<img border="0" src="images/furutadyn1.JPG" width="276" height="214"></p>
			<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal"><b><a name="Cartpole">2.1.2. Cartpole</a></b></p>
			<p align="center">
			<img border="0" src="images/cartpole.JPG" width="294" height="176"></p>
			<p class="MsoNormal" style="text-align:justify">Applying Newton&#8217;s
			<!--[if gte mso 9]>second<![endif]-->&nbsp;law 
			at the center of mass of the pendulum along the horizontal and 
			vertical components, yields:</p>
			<p align="justify">
			<img border="0" src="images/cartdyn1.JPG" width="372" height="145"></p>
			<p align="justify">
			<img border="0" src="images/cartdyn3.JPG" width="875" height="334"></p>
			<p>
			<img border="0" src="images/cartdyn2.JPG" width="448" height="272"></p>
			<p><span style="line-height: 115%; font-family: Times New Roman">The 
			Motor Dynamics equations included:</span></p>
			<p>
			<img border="0" src="images/cartdyn4.JPG" width="185" height="56"></p>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal"><span style="font-family: Times New Roman; font-style: italic">
			Parameters description:</span><font face="Times New Roman"><br>
			</font>
			<img border="0" src="images/cartdyn5.JPG" width="245" height="266"></p>
			<p align="right"><i><a href="#.">Back Top</a></i></p>
	<hr>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal"><b><a name="Acrobot_and_Pendubot">2.1.3. Acrobot and Pendubot</a></b></p>
		</blockquote>
		<p align="center">
		<img border="0" src="images/AcroPendu.JPG" width="596" height="229"></p>
		<p align="left">&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
		Pendubot&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
		Acrobot</p>
		<blockquote>
			<p>
			<img border="0" src="images/acropendydyn.JPG" width="864" height="379"></p>
			<p>Note that:</p>
			<p>
			<img border="0" src="images/acropenddyn1.JPG" width="731" height="410"></p>
			<p><span style="line-height: 115%; font-family: Times New Roman">The 
			Motor Dynamics equations included:</span></p>
			<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
			<img border="0" src="images/acropendyn2.JPG" width="651" height="68"></p>
			<p><span style="font-family: Times New Roman; font-style: italic">
			Parameters description:</span><span style="font-family: Arial Narrow">&nbsp;</span></p>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal"><span style="font-family: Arial Narrow">&nbsp;</span><img border="0" src="images/acropendyn3.JPG" width="351" height="406"></p>
			<p align="right"><i><a href="#.">Back Top</a></i></p><hr>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal" align="left"><b><a name="Inertia_Wheel">2.1.4. Inertia Wheel</a></b></p>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal" align="center">
			<img border="0" src="images/wheel.JPG" width="401" height="417"></p>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal" align="center">
			&nbsp;</p>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal" align="left">
			<img border="0" src="images/wheeldyn.JPG" width="529" height="505"><br>
			<br>
			<img border="0" src="images/wheeldyn1.JPG" width="355" height="497"></p>
			<p class="MsoNormal" style="margin-bottom:0in;margin-bottom:.0001pt;line-height:
normal" align="right"><i><a href="#.">Back Top</a></i><br>
&nbsp;</p><hr noshade>
		</blockquote>
	</blockquote>
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